Development of a dynamic model of transients in mechanical systems using argument-functions

Abstract

There are a number of applied problems in which it is necessary to take into account the dynamic component of the process or phenomenon including the fact that the load is applied not instantaneously but in time. For example, in continuous rolling, such combinations of mechanical systems appear in which action transfer from one rolling stand to another via the strip proceeds with some delay affecting transient processes and the strip gripping capability in the adjacent stands of the continuous mill. The strip between the mill stands is in an elastic state. When the rolls start acting on it during the bite in the next stand, they transfer disturbance to the strip in a form of oscillations or in a form of a stationary action. The aim of this research was to expand the application field of the obtained solutions to satisfy boundary and initial conditions formulated by applied production problems. The wave problem was considered as the process of propagation of the initial deviation and initial velocity. On the basis of the method, the essence of which is the use of argument-functions, solution of dynamic linear and spatial problems of the elasticity theory was shown. In the course of the study, conditions for existence of new solutions for the wave problem, which are limited by the boundary conditions of various processes were shown. The initial differential equations and boundary conditions determine the type of differential equations for the argument-functions that close the solution. Argument-functions can be restricted by the Cauchy-Riemann relations and the corresponding differential invariants on the one hand and the differential relationships which result in that the argument-functions are the same for adjacent coordinate-time dependencies on the other hand. Besides, analytical dependences on the parameters entering into the d’Alembert formula were obtained.